abelian variety (original) (raw)

Definition 1.

This extremely terse definition needs some further explanation.

This implies that for every ring R, the R-points of an abelian variety form an abelian group.

Proposition 2.

An abelian variety is projective.

If C is a curve, then the Jacobian of C is an abelian variety. This example motivated the development of the theory of abelian varieties, and many properties of curves are best understood by looking at the Jacobian.

If E is an elliptic curve, then E is an abelian variety (and in fact E is naturally isomorphic to its Jacobian).